Post on 30-Jan-2016
description
Relativistic Models of Quasielastic Electron and Neutrino-Nucleus
Scattering
Carlotta GiustiUniversity and INFN Pavia
in collaboration with: A. Meucci (Pavia) F.D. Pacati (Pavia)
J.A. Caballero (Sevilla) J.M. Udías (Madrid)
XXVIII International Workshop in Nuclear Theory, Rila Mountains June 21st-27th 2009
nuclear response to the electroweak probe
QE-peak
QE-peak dominated by one-nucleon knockout
QE e-nucleus scattering
both e’ and N detected one-nucleon-knockout (e,e’p)
(A-1) is a discrete eigenstate n exclusive (e,e’p)
QE e-nucleus scattering
both e’ and N detected one-nucleon-knockout (e,e’p)
(A-1) is a discrete eigenstate n exclusive (e,e’p)
only e’ detected inclusive (e,e’)
QE e-nucleus scattering
NC
CC
both e’ and N detected one-nucleon knockout (e,e’p)
(A-1) is a discrete eigenstate n exclusive (e,e’p)
only e’ detected inclusive (e,e’)
QE e-nucleus scattering
QE -nucleus scattering
only N detected semi-inclusive NC and CC
NC
CC
both e’ and N detected one-nucleon knockout (e,e’p)
(A-1) is a discrete eigenstate n exclusive (e,e’p)
only e’ detected inclusive (e,e’)
QE e-nucleus scattering
QE -nucleus scattering
only N detected semi-inclusive NC and CC
only final lepton detected inclusive CC
NC
CC
both e’ and N detected one-nucleon knockout (e,e’p)
(A-1) is a discrete eigenstate n exclusive (e,e’p)
only e’ detected inclusive (e,e’)
QE e-nucleus scattering
QE -nucleus scattering
electron electron scatteringscattering
NCNC
neutrino neutrino scatteringscattering
CCCC
PVESPVES
electron electron scatteringscattering
NCNC
neutrino neutrino scatteringscattering
CCCC
PVESPVES
kinkin factorfactor
electron electron scatteringscattering
NCNC
neutrino neutrino scatteringscattering
CCCC
PVESPVES
lepton tensor contains lepton lepton tensor contains lepton
kinematicskinematics
electron electron scatteringscattering
NCNC
neutrino neutrino scatteringscattering
CCCC
PVESPVES
hadron hadron tensortensor
e’
eq
p n
0
Direct knockout DWIA (e,e’p)
exclusive reaction: n
DKO mechanism: the probe interacts through a one-body current with one nucleon which is then emitted the remaining nucleons are spectators
|i >
|f >
e’
eq
p n
0
Direct knockout DWIA (e,e’p)
|i >
|f >
j one-body nuclear current
(-) = < n|f> s.p. scattering w.f. H+
(+Em)
n = <n|0> one-nucleon overlap H(-Em)
n spectroscopic factor
(-) and consistently derived as eigenfunctions of a Feshbach-type optical model Hamiltonian
phenomenological ingredients used in the calculations for (-) and
rel RDWIA
nonrel DWIA
RDWIA diff opt.pot.
RDWIA: (e,e’p) comparison to data
NIKHEF parallel kin e=520 MeV Tp = 90 MeV
16O(e,e’p)
n = 0.7
n =0.65
JLab (,q) const kin e=2445 MeV =439 MeV Tp= 435 MeV
n = 0.7
k’
kq
N n
0
RDWIA: NC and CC –nucleus scattering
|i >
|f >
transition amplitudes calculated with the same model used for (e,e’p)
the same phenomenological ingredients are used for (-) and
j one-body nuclear weak current
One-body nuclear weak current
NC
K anomalous part of the magnetic moment
CC
induced pseudoscalar form factor
The axial form factor
possible strange-quark contribution
CC
NC
One-body nuclear weak current
One-body nuclear weak current
The weak isovector Dirac and Pauli FF are related to the Dirac and Pauli elm FF by the CVC hypothesis
strange FF
CC
NC
–nucleus scattering
only the outgoing nucleon is detected: semi inclusive process
cross section integrated over the energy and angle of the outgoing lepton
integration over the angle of the outgoing nucleon
final nuclear state is not determined : sum over n
pure Shell Model description: n one-hole states in the target with an unitary spectral strengthn over all occupied states in the SM: all the nucleons are included but correlations are neglectedthe cross section for the -nucleus scattering where one nucleon is detected is obtained from the sum of all the integrated one-nucleon knockout channelsFSI are described by a complex optical potential with an imaginary absorptive part
calculationscalculations
RPWIARPWIA
RDWIARDWIA
RPWIARPWIA
RDWIARDWIA
CC CC
NCNC
FSIFSI
FSIFSI
the imaginary part of the the imaginary part of the optical potential gives an optical potential gives an absorption that reduces the absorption that reduces the calculated cross sectionscalculated cross sections
FSIFSI
FSI for the inclusive scattering : Green’s Function Approach
(e,e’) nonrelativistic(e,e’) nonrelativistic
F. Capuzzi, C. Giusti, F.D. Pacati, Nucl. Phys. A 524 (1991) 281F. Capuzzi, C. Giusti, F.D. Pacati, Nucl. Phys. A 524 (1991) 281
F. Capuzzi, C. Giusti, F.D. Pacati, D.N. Kadrev Ann. Phys. 317 (2005) 492F. Capuzzi, C. Giusti, F.D. Pacati, D.N. Kadrev Ann. Phys. 317 (2005) 492 (AS CORR)(AS CORR)
(e,e’) relativistic(e,e’) relativistic
A.A. MeucciMeucci, , F. Capuzzi, C. Giusti, F.D. Pacati, PRC (2003) 67 054601 F. Capuzzi, C. Giusti, F.D. Pacati, PRC (2003) 67 054601
A. Meucci, C. Giusti, F.D. Pacati Nucl. Phys. A 756 (2005) 359A. Meucci, C. Giusti, F.D. Pacati Nucl. Phys. A 756 (2005) 359 (PVES)(PVES)
CC relativisticCC relativistic
A. Meucci, C. Giusti, F.D. Pacati Nucl. Phys. A739 (2004) 277A. Meucci, C. Giusti, F.D. Pacati Nucl. Phys. A739 (2004) 277
FSI for the inclusive scattering : Green’s Function Approach
the components of the inclusive response are expressed in terms of thethe components of the inclusive response are expressed in terms of the Green’s operatorsGreen’s operators
under suitable approximations can be written in terms of theunder suitable approximations can be written in terms of the s.p. optical s.p. optical model Green’s function model Green’s function
the explicit calculation of the s.p. Green’s function can be avoided by its the explicit calculation of the s.p. Green’s function can be avoided by its spectral representation which is based on a biorthogonal expansion in spectral representation which is based on a biorthogonal expansion in terms of aterms of a non Herm optical potential V and Vnon Herm optical potential V and V++
matrix elements similar to RDWIA matrix elements similar to RDWIA
scattering states eigenfunctions of V and Vscattering states eigenfunctions of V and V++ (absorption and gain of (absorption and gain of flux): flux): the imaginary part redistributes the flux and the total flux is the imaginary part redistributes the flux and the total flux is conservedconserved
consistent treatmentconsistent treatment of FSI in the exclusive and in the inclusive of FSI in the exclusive and in the inclusive scatteringscattering
FSI for the inclusive scattering : Green’s Function Approach
interference between different channels
FSI for the inclusive scattering : Green’s Function Approach
eigenfunctions of V and V+
FSI for the inclusive scattering : Green’s Function Approach
Flux redistributed and conserved
The imaginary part of the optical potential is responsible for the redistribution of the flux among the different channels
gain of flux loss of flux
FSI for the inclusive scattering : Green’s Function Approach
For a real optical potential V=V+ the second term vanishes and the nuclear response is given by the sum of all the integrated one-
nucleon knockout processes (without absorption)
gain of flux loss of flux
16O(e,e’)
data from Frascati NPA 602 405 (1996)
Green’s function approach GF
A. Meucci, F. Capuzzi, C. Giusti, F.D. Pacati, PRC 67 (2003) 054601
16O(e,e’)
data from Frascati NPA 602 405 (1996)
e = 1080 MeV
=320
e = 1200 MeV
=320
RPWIARPWIA
GFGF
1NKO1NKO
FSI FSI
RPWIARPWIA
GFGF
rROPrROP
1NKO1NKO
GFGF
FSI FSI FSI FSI
PAVIA MADRID-SEVILLA
COMPARISON OF RELATIVISTIC MODELS
consistency of numerical results
comparison of different descriptions of FSI
12C(e,e’)
RPWIARPWIA
rROPrROP
GF1GF1
GF2GF2
RMFRMF
e = 1 GeV
relativistic models
FSI FSI
Relativistic Mean Field: same real energy-independent potential of bound states
Orthogonalization, fulfills dispersion relations and maintains the continuity equation
12C(e,e’)
GF1GF1
GF2GF2
RMFRMF
e = 1 GeV
relativistic models
q=500 MeV/cq=500 MeV/c
q=800 MeV/cq=800 MeV/c
q=1000 MeV/cq=1000 MeV/c
FSI FSI
12C(e,e’)
GF1GF1
GF2GF2
RMFRMF
relativistic models
GF RMF
SCALING PROPERTIES
SCALING FUNCTION
Scaling properties of the electron scattering data Scaling properties of the electron scattering data
At sufficiently high q the At sufficiently high q the scaling functionscaling function
depends only upon one kinematical variable (scaling variable) depends only upon one kinematical variable (scaling variable)
(SCALING OF I KIND)(SCALING OF I KIND)
is the same for all nucleiis the same for all nuclei
(SCALING OF II KIND)(SCALING OF II KIND)
I+II SUPERSCALING I+II SUPERSCALING
Scaling variable (QE)Scaling variable (QE)
+ (-) for + (-) for lower (higher) than the QEP, where lower (higher) than the QEP, where =0 =0
Reasonable scaling of I kind at the left of QEPReasonable scaling of I kind at the left of QEP
Excellent scaling of II kind in the same regionExcellent scaling of II kind in the same region
Breaking of scaling particularly of I kind at the Breaking of scaling particularly of I kind at the right of QEP (effects beyond IA) right of QEP (effects beyond IA)
The longitudinal contribution superscalesThe longitudinal contribution superscales
ffQEQE extracted from the data extracted from the data
Experimental QE superscaling functionExperimental QE superscaling function
M.B. Barbaro, J.E. Amaro, J.A. Caballero, T.W. Donnelly, A. Molinari, and I. Sick, Nucl. Phys Proc. Suppl 155 (2006) 257
The properties of the experimental scaling function should be The properties of the experimental scaling function should be accounted for by microscopic calculationsaccounted for by microscopic calculations
The asymmetric shape of fThe asymmetric shape of fQEQE should be explained should be explained
The scaling properties of different models can be verified The scaling properties of different models can be verified
The associated scaling functions compared with the experimental The associated scaling functions compared with the experimental ffQEQE
SCALING FUNCTION
QE SUPERSCALING FUNCTION: RFGQE SUPERSCALING FUNCTION: RFG
Relativistic Fermi Gas
J.A. Caballero J.E. Amaro, M.B. Barbaro, T.W. Donnelly, C. Maieron, and J.M.
Udias PRL 95 (2005) 252502
only RMF gives only RMF gives an asymmetric an asymmetric
shapeshape
QE SUPERSCALING FUNCTION: RPWIA, rROP, RMFQE SUPERSCALING FUNCTION: RPWIA, rROP, RMF
QE SCALING FUNCTION: GF, RMFQE SCALING FUNCTION: GF, RMF
GF1GF1
GF2GF2
RMFRMF
asymmetric asymmetric shapeshape
q=500 MeV/cq=500 MeV/c
q=800 MeV/cq=800 MeV/c
q=1000 MeV/cq=1000 MeV/c
Analysis first-kind scaling : GF RMFAnalysis first-kind scaling : GF RMF
GF1GF1
GF2GF2
RMFRMF
q=500 MeV/cq=500 MeV/c
q=800 MeV/cq=800 MeV/c
q=1000 q=1000 MeV/cMeV/c
RMF GF
DIFFERENT DESCRIPTIONS OF FSI
real energy-independent MF reproduces nuclear saturation properties, purely nucleonic contribution, no information from scattering reactions explicitly incorporated
complex energy-dependent phenomenological OP fitted to elastic p-A scattering information from scattering reactions
the imaginary part includes the overall effect of inelastic channels not univocally determined from elastic phenomenology
different OP reproduce elastic p-A scattering but can give different predictions for non elastic observables
RESULTS :RESULTS :
GF1, GF2, RMF similar results for the cross section and QE scaling GF1, GF2, RMF similar results for the cross section and QE scaling function for q=500-700 MeV/c, differences increase increasing qfunction for q=500-700 MeV/c, differences increase increasing q
WHY ?WHY ?
At higher q and energies the phenomenological OP can include At higher q and energies the phenomenological OP can include contributions from non nucleonic d.o.f. (flux lost into inelastic contributions from non nucleonic d.o.f. (flux lost into inelastic excitation with or without real excitation with or without real production) production)
GF can give better description of (e,e’) experimental c.s. at higher GF can give better description of (e,e’) experimental c.s. at higher q, where QE and q, where QE and peak approach and overlap peak approach and overlap
Non nucleonic contributions in the OP break scaling, they should Non nucleonic contributions in the OP break scaling, they should not be included in the QE longitudinal scaling function (purely not be included in the QE longitudinal scaling function (purely nucleonic) nucleonic)
COMPARISON RMF-GF
FUTURE WORK…..
deeper understanding of the differences deeper understanding of the differences
disentangle nucleonic and non nucleonic contributions in the OP, disentangle nucleonic and non nucleonic contributions in the OP, extract purely nucleonic OP, use it in the GF approach and extract purely nucleonic OP, use it in the GF approach and compare the results with the QE experimental scaling functioncompare the results with the QE experimental scaling function
extend the comparison to different situationsextend the comparison to different situations, neutrino scattering…